Preservation of the residual classes numbers by polynomials

Abstract

Let K be a global field and let O K,S be the ring of S-integers of K for some finite set S of primes of K. We prove that whatever the infinite subset E C O K,S and the polynomial f(X) ∈ K[X], the subsets E and f(E) have the same number of residual classes modulo m for almost all maximal ideals m of O K,S if and only if deg(f) = 1 when the characteristic of K is 0 and f(X) = g(X pk ) for some integer k and some polynomial g with deg(g) = 1 when the characteristic of K is p > 0.